... taken.¹

It is easy to extend the analysis to the infinite Fourier space, $k_{max} = \infty$. In this case, the full joint PDF would still have to be defined as a $N \to \infty$ limit of an $N$-particle PDF, but this limit would have to be taken in such a way that both $k_{max}$ and the density of the Fourier modes tend to infinity simultaneously.

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... symmetries.²

Some additional symmetries involving permutations of the upper and lower indices arise, e.g., in solids due to the fact that nonlinearity is purely due to the potential energy which is a function of the displacement but not the rate of the displacement. Refs. [15,16,13] imposed such symmetries which immediately rule out the capillary, internal and other waves in fluids for which such properties do not hold. Additional symmetries also arise if the action variable is a Fourier transform of a real quantity, e.g., in the Rossby waves [9].

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... scales.³

Hereafter we omit superscript ${(N)}$ in the $N$-particle objects if it does not lead to a confusion.

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